Economics 101
Answer to Homework #4
Spring 2009
Due 3/31/2009 at beginning of lecture
Directions: The homework will be collected in a box before the lecture. Please place
your name, TA name and section number on top of the homework (legibly). Make
sure you write your name as it appears on your ID so that you can receive the correct
grade. Please remember the section number for the section you are registered,
because you will need that number when you submit exams and homework. Late
homework will not be accepted so make plans ahead of time. Please show your work.
Good luck!
Question 1
Staley is at a carnival. The following table gives the total utility that Staley gets from
eating bratwursts at the carnival.
Number of
Total Utility
Marginal
Bratwursts
Utility
0
0
--1
6
6
2
12
6
3
17
5
4
19
2
5
20
1
6
19
-1
1) Calculate the marginal utility of eating each unit of bratwurst.
2) If the bratwursts are free, how many bratwursts would Staley choose to eat?
Staley will eat 5 Bratwursts.
Suppose there is also free ice cream at the carnival. The following table gives the
marginal utility of eating ice cream. Calculate the total utility of eating each unit of
ice cream.
Ice Cream
Total Utility
Marginal
Utility
0
0
---1
5
5
2
9
4
3
12
3
4
14
2
5
15
1
6
15
0
3) Suppose now that Staley has to pay for the bratwursts and ice cream. Each
bratwurst cost $2 and each ice cream costs $4. Staley’s income equals $16.
How many bratwursts and ice cream would Staley consume?
Staley will consume 2 units of ice cream and 4 units of bratwursts.
4) Suppose the price of ice cream decreases to $2. Holding everything else
constant, how many bratwursts and ice cream would Staley consume?
Staley will consume 4 bratwursts and 4 units of ice cream.
5) Assume Staley’s demand function for ice cream is linear. Using the results
from part 3) and part 4), derive Staley’s demand function.
Change in P/Change in Q=(4-2)/(2-4)=-1.
P-2=-1(Q-4) => P=-Q+6
Q2: Budget Lines and Indifference Curves
A. Suppose Tayler has an income of $480 to buy two goods: pizza and textbooks. The price
of a slice of pizza is $2, and the price of a textbook is $80.
1) Draw Tayler’s budget line (BL1) given her income is $480. (Put textbooks on the
X-axis and pizza on the Y-axis.) Assume Tayler’s utility function is U(x,y)=xy2
(x is the purchase of the number of textbooks and y is the purchase of the
number of slices of pizza). Tayler’s marginal utility of consuming a slice of pizza
is 2xy. Tayler’s marginal utility of consuming a textbook is y2.
pizza
240
200
160
120
80
40
BL1
0
0
1
2
3
4
5
6
textbook
7
2) Derive Tayler’s Marginal Rate of Substitution of textbooks for pizza.
MRS=MUx/MUy=y2/(2xy)=y/(2x)
3) Derive Tayler’s budget line.
2y+80x=480
4) Find the optimal consumption point. What is Tayler’s utility? On your graph
draw the indifference curve (IC1).
MRS=Px/Py => y/(2x)=80/2 => y=80x.
Substitute into budget line: 2(80x)+80x=480 => 240x=480 => x=2, y=160.
Utility=2(1602)=51200
pizza
240
200
160
(2,160)
120
80
IC1
40
BL1
0
0
1
2
3
4
5
textbook
6
7
5) Suppose the price of textbooks decreases to $40. Find the optimal consumption point
given this new price for textbooks. Draw the new budget line (BL2) and indifference
curve (IC2) on your graph. Label the budget line and indifference curve clearly.
MRS=Px/Py => y/(2x)=40/2 => y=40x.
Substitute into budget line: 2(40x)+40x=480 => 120x=480 => x=4, y=160.
pizza
240
200
160
(2,160)
(4,160)
120
80
IC2
IC1
40
BL1
0
0
1
2
3
4
5
6
BL2
7
8
9
10
11
12
textbook
13
6) Given the price change described in part (4), if Tayler consumes optimally,
how much income would be required for Tayler to obtain the same level of
utility that she had in part (3)? After you calculate this compensated level of
income, draw an imaginary budget line (BL3) using this new income and the
new prices on your graph.
MRS=Px/Py => y/(2x)=40/2 => y=40x.
To reach 51200 utility. xy2=51200 => 1600x3=51200 => x=321/3≈3.175.
Y=40(321/3)≈126.992
Income required = 2y+40x=2*40(321/3)+40(321/3)=120(321/3)≈380.976
Imaginary budget line is 2y+40x=120(321/3)
pizza
240
200
160
(2,160)
(4,160)
(3.2,127)
120
80
IC2
IC1
40
BL1
0
0
1
2
3
4
5
6
BL3
7
8
BL2
9
10
11
12
textbook
13
7) What is the substitution and income effect in textbook consumption due to the
change in the price of textbooks? Illustrate the effect on your graph.
Substitution effect is 321/3-2≈1.175 textbooks.
Income effect is 4-321/3≈0.825 textbooks.
pizza
240
200
160
(2,160)
(4,160)
(3.2,127)
120
80
IC2
IC1
40
substitution income
0
0
1
2
3
4
BL1
5
6
BL3
7
8
BL2
9
10
11
12
textbook
13
B. Sarah loves musicals. Sarah receives 15 units of utility for every Les Miserables
performance she goes go. Sarah receives 10 units of utility for every Phantom of
the Opera performance she goes to. Suppose Sarah’s income is $300. The price of
either shows is $100 each. Assume that the utility from consumption of an
additional unit of either good is constant.
1) Draw Sarah’s budget line (BL1) given her income is $300. (Measure Phantom
of the Opera on the Y-axis and Les Miserables on the X-axis.)
Phantom of the Opera
3
2
1
BL1
0
0
1
2
Les Miserables
3
2) Please find the utility maximization point and draw an indifference curve (IC1)
through the utility maximization point on your graph.
Sarah will watch 3 Les Miserables performances and 0 Phantom of the Opera.
Phantom of the Opera
IC1
4
3
2
1
0
0
BL1
1
2
(3,0)
Les Miserables
3
3) Suppose the price of a Les Miserables performance increases to $200, draw
Sarah’s new budget line (BL2) and find her new utility maximization
consumption bundle on your graph.
Sarah will watch 3 Phantom of the Opera performances and 0 Les Miserables.
Phantom of the Opera
IC1
4
3 (0,3)
2
1
BL2
0
0
1
IC2
BL1
2
Les Miserables
(3,0)
3
4) Draw an imaginary budget line (BL3) parallel to the new budget line (BL2)
and make it touch the initial indifference curve (IC1) at the lowest possible
income level. Calculate and show the income and substitution effects in the
consumption of Les Miserables performances due to the price increase.
Substitution effect is -3 Les Miserables performancess.
Income effect is 0 Les Miserables performances.
Phantom of the Opera
(0,4.5)
IC1
substitution
4
3 (0,3)
2
1
BL2
0
0
1
IC2 BL3 BL1
2
(3,0)
Les Miserables
3
C. Amanda is buying orange juice and vodka to mix drinks. She wants 1 bottle of
vodka for every carton of orange juice. She does not like orange juice by itself and
she does not like drinking straight vodka. The price of a cartoon of orange juice is
$2, and the price of a bottle of vodka is $10.
1) Draw Amanda’s budget line (BL1) given her income is $36. (Put vodka on the
X-axis and orange juice on the Y-axis.)
Orange Juice
18
16
14
12
10
8
6
4
2
0
BL1
Vodka
0
1
2
3
2) Please find the utility maximization point and draw an indifference curve (IC1)
through the utility maximization point.
Amanda will purchase 3 bottles of vodka and 3 cartons of orange juice.
Orange Juice
IC1
18
16
14
12
10
8
6
4
(3,3)
2
0
BL1
Vodka
0
1
2
3
4
3) Suppose the price of a bottle of vodka increases to $16, draw Amanda’s new
budget line (BL2) and find her new utility maximization consumption bundle.
Amanda will purchase 2 bottles of vodka and 2 cartons of orange juice.
Orange Juice
IC2
IC1
18
16
14
12
10
8
6
4
(3,3)
2
0
(2,2)
0
1
BL1
BL2
2
Vodka
3
4
4) Draw an imaginary budget line (BL3) parallel to the new budget line (BL2) and
make it touch the initial indifference curve (IC1) at the lowest income level
possible. Calculate and show the income and substitution effects on the
consumption of vodka due to the price change in vodka.
Substitution effect is 0 vodka.
Income effect is -1 vodka.
Orange Juice
26
24
BL3
22
20
IC2
IC1
18
16
income
14
12
10
8
6
4
(3,3)
2
0
(2,2)
0
1
BL1
BL2
2
3
Vodka
4
Q3: Production Cost I
The following table gives cost information for a firm. Assume that labor is paid a
constant wage and capital is paid a constant price, i.e., our firm is a price-taker both in
the labor and capital markets.
L
K
Q
VC
FC
TC
AVC
AFC
ATC
MC
MPL
0
5
0
0
100
100
---
---
---
---
---
1
5
225
100
100
200
4/9
4/9
8/9
4/9
225
2
5
600
200
100
300
1/3
1/6
1/2
4/15
375
3
5
900
300
100
400
1/3
1/9
4/9
1/3
300
4
5
1140
400
100
500
20/57
5/57
25/57
5/12
240
5
5
1340
500
100
600
25/67
5/67
30/67
1/2
200
6
5
1400
600
100
700
3/7
1/14
1/2
5/3
60
1) Suppose output is equal to zero. Why does the firm still incur costs in the short
run when output is equal to zero? What is the fixed cost of this firm? What is
the price of a unit of capital?
Fixed cost is always incurred. Fixed cost is 100. Price per unit of capital is
$20.
2) What is the wage rate?
Wage is $100
3) Please complete the above table with specific numbers.
4) At what level of output and labor usage is AVC at its minimum?
Output =900, labor = 3
5) At what level of output and labor usage is ATC at its minimum?
Output =1140, labor = 4
6) At what level of labor usage does the law of diminishing returns first occur?
Occurs at the 3 unit of labor.
7) According to the table, what is the minimum price for the firm to be willing to
hire the fourth unit of labor? According to the table, what is the minimum
price for the firm to be willing to hire the fifth unit of labor? What is the price
range of the product if the optimal number of labor hired by the firm is 4?
Price is at least 5/12 and strictly less than ½.
Q4: Perfect Competition
Suppose a perfectly competitive firm’s cost function is C(q)=4q2+16. Marginal cost
for the firm is given by MC=8q.
1) Find equations for variable cost, fixed cost, average total cost, average
variable cost and average fixed cost for this firm. Illustrate on a graph the
firm’s average variable cost curve, average total cost curve, and marginal cost
curve.
VC=4q2, FC=16, ATC=(4q2+16)/q, AVC=4q, AFC=16/q
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
$
ATC
MC
AVC
Q
0
1
2
3
4
5
6
7
8
9
2) Find the outputs that minimize average total cost, average variable cost and
average fixed cost.
ATC is minimized at (4q2+16)/q=8q => (4q2+16)=8q2 => 4=q2 => q=2.
AVC is minimized at q=0.
ATC is minimized as q approaches infinity.
3) What is the shutdown price for this firm?
Shutdown price is 0.
4) What is the breakeven price for this firm?
Breakeven price is 16.
5) If the price of the output is $32, how many units will this firm produce? What
is the firm’s profit?
32=8q => q=4.
Profit = 4*32 – 4(42)-16= 48
6) Suppose market demand for this good is given as P=300-Q and all firms in the
market are identical. How many firms are active in the market in the long run?
P=16. Q=300-16=284. n=284/2=142.